statistical and dynamical aspects of mesoscopic systems

Preface
One of the most significant developments in physics in recent years con-
cerns mesoscopic systems, a subfield of condensed matter physics which has
achieved proper identity. The main objective of mesoscopic physics is to un-
derstand the physical properties of systems that are not as small as single
atoms, but small enough that properties can differ significantly from those of
a large piece of material. This field is not only of fundamental interest in its
own right, but it also offers the possibility of implementing new generations
of high-performance nano-scale electronic and mechanical devices. In fact,
interest in this field has been initiated at the request of modern electronics
which demands the development of more and more reduced structures. Un-
derstanding the unusual properties these structures possess requires collabo-
ration between disparate disciplines. The future development of this promis-
ing field depends on finding solutions to a series of fundamental problems
where, due to the inherent complexity of the devices, statistical mechanics
may play a very significant role. In fact, many of the techniques utilized in
the analysis and characterization of these systems have been borrowed from
that discipline.
Motivated by these features, we have compiled this new edition of the Sit-
ges Conference. We have given a general overview of the field including top-
ics such as quantum chaos, random systems and localization, quantum dots,
noise and fluctuations, mesoscopic optics, quantum computation, quantum
transport in nanostructures, time-dependent phenomena, and driven tunnel-
ing, among others.
The Conference was the first of a series of two Euroconferences focusing
on the topic Nonlinear Phenomena in Classical and Quantum Systems.It
was sponsored by CEE (Euroconference) and by institutions who generously
provided financial support: DGCYT of the Spanish Government, CIRIT of
the Generalitat of Catalunya, the European Physical Society, Universitat de
Barcelona and Universidad Carlos III de Madrid. It was distinguished by the
European Physical Society as a Europhysics Conference. The city of Sitges

allowed us, as usual, to use the Palau Maricel as the lecture hall.
VI Preface
Finally, we are also very grateful to all those who collaborated in the
organization of the event, Profs. F. Guinea and F. Sols, Drs. A. P´erez-Madrid
and O. Bulashenko, as well as M. Gonz´alez, T. Alarc´on and I. Santamar´ıa-
Holek.
Barcelona, February 2000 The Editors
Contents
Part I Quantum Dots
Thermopower in Quantum Dots
K.A. Matveev 3
Kondo Effect in Quantum Dots
L.I. Glazman, F.W.J. Hekking, and A.I. Larkin 16
Interpolative Method for Transport Properties
of Quantum Dots in the Kondo Regime
A.L. Yeyati, A. Mart´ın-Rodero, and F. Flores 27
A New Tool for Studying Phase Coherent Phenomena
in Quantum Dots
R.H. Blick, A.W. Holleitner, H. Qin, F. Simmel, A.V. Ustinov,
K. Eberl, and J.P. Kotthaus 35
Part II Quantum Chaos
Quantum Chaos and Spectral Transitions in the Kicked
Harper Model
K. Kruse, R. Ketzmerick, and T. Geisel 47
Quantum Chaos Effects in Mechanical Wave Systems
S.W. Teitsworth 62
Magnetoconductance in Chaotic Quantum Billiards
E. Louis and J.A. Verg´es 69
VI II Contents
Part III Time-Dependent Phenomena

Shot Noise Induced Charge and Potential Fluctuations
of Edge States in Proximity of a Gate
M. B¨uttiker 81
Shot-Noise in Non-Degenerate Semiconductors
with Energy-Dependent Elastic Scattering
H. Schomerus, E.G. Mishchenko, and C.W.J. Beenakker 96
Transport and Noise of Entangled Electrons
E.V. Sukhorukov, D. Loss, and G. Burkard 105
Shot Noise Suppression in Metallic Quantum Point Contacts
H.E. van den Brom and J.M. van Ruitenbeek 114
Part IV Driven Tunneling
Driven Tunneling: Chaos and Decoherence
P. H¨anggi, S. Kohler, and T. Dittrich 125
A Fermi Pump
M. Wagner and F. Sols 158
Part V Transport in Semiconductor Superlattices
Transport in Semiconductor Superlattices: From Quantum
Kinetics to Terahertz-Photon Detectors
A.P. Jauho, A. Wacker, and A.A. Ignatov 171
Current Self-Oscillations and Chaos
in Semiconductor Superlattices
H.T. Grahn 193
Part VI Spin Properties
Spintronic Spin Accumulation and Thermodynamics
A.H. MacDonald 211
Mesoscopic Spin Quantum Coherence
J.M. Hernandez, J. Tejada, E. del Barco, N. Vernier, G. Bellessa,
and E. Chudnovsky 226
Contents IX
Part VII Random Systems and Localization

Numerical-Scaling Study of the Statistics of Energy Levels
at the Anderson Transition
I.Kh. Zharekeshev and B. Kramer 237
Multiple Light Scattering in Nematic Liquid Crystals
D.S. Wiersma, A. Muzzi, M. Colocci, and R. Righini 252
Two Interacting Particles
in a Two-Dimensional Random Potential
M. Ortu˜no and E. Cuevas 263
Part VIII Mesoscopic Superconductors, Nanotubes
and Atomic Chains
Paramagnetic Meissner Effect
in Mesoscopic Superconductors
J.J. Palacios 273
Novel 0D Devices: Carbon-Nanotube Quantum Dots
L. Chico, M.P. L´opez Sancho, and M.C. Mu˜noz 281
Atomic-Size Conductors
N. Agra¨ıt 290
Appendix I Contributions Presented as Posters
Observation of Shell Structure in Sodium Nanowires
A.I. Yanson, I.K. Yanson, and J.M. van Ruitenbeek 305
Strong Charge Fluctuations in the Single-Electron Box:
A Quantum Monte Carlo Analysis
C.P. Herrero, G. Sch¨on, and A.D. Zaikin 306
Double Quantum Dots as Detectors of High-Frequency
Quantum Noise in Mesoscopic Conductors
R. Aguado and L.P. Kouwenhoven 307
Large Wigner Molecules and Quantum Dots
C.E. Creffield, W. H¨ausler, J.H. Jefferson, and S. Sarkar 308
X Contents
Fundamental Problems for Universal Quantum Computers

T.D. Kieu and M. Danos 309
Kondo Photo-Assisted Transport in Quantum Dots
R. L´opez, G. Platero, R. Aguado, and C. Tejedor 310
Shot Noise and Coherent Multiple Charge Transfer
in Superconducting Quantum Point-Contacts
J.C. Cuevas, A. Mart´ın-Rodero, and A.L. Yeyati 311
Evidence for Ising Ferromagnetism and First-Order Phase
Transitions in the Two-Dimensional Electron Gas
V. Piazza, V. Pellegrini, F. Beltram, W. Wegscheider, M. Bichler,
T. Jungwirth, and A.H. MacDonald 312
Mechanical Properties
of Metallic One-Atom Quantum Point Contacts
G.R. Bollinger, N. Agra¨ıt, and S. Vieira 314
Nanosized Superconducting Constrictions
in High Magnetic Fields
H. Suderow, E. Bascones, W. Belzig, S. Vieira, and F. Guinea 315
Interaction-Induced Dephasing
in Disordered Electron Systems
S. Sharov and F. Sols 316
Resonant Tunneling Through Three Quantum Dots
with Interdot Repulsion
M.R. Wegewijs, Yu.V. Nazarov, and S.A. Gurvitz 317
Spin-Isospin Textures in Quantum Hall Bilayers
at Filling Factor ν =2
B. Paredes, C. Tejedor, L. Brey, and L. Mart´ın-Moreno 318
Hall Resistance of a Two-Dimensional Electron Gas
in the Presence of Magnetic Clusters
with Large Perpendicular Magnetization
J. Reijniers, A. Matulis, and F.M. Peeters 319
Superconductivity Under Magnetic Fields

in Nanobridges of Lead
H. Suderow, A. Izquierdo, E. Bascones, F. Guinea, and S. Vieira 320
Contents XI
Effect of the Measurement on the Decay Rate
of a Quantum System
B. Elattari and S. Gurvitz 321
Statistics of Intensities in Surface Disordered Waveguides
A. Garc´ıa-Mart´ın, J.J. S´aenz, and M. Nieto-Vesperinas 322
Optical Transmission Through Strong Scattering
and Highly Polydisperse Media
J.G. Rivas, R. Sprik, C.M. Soukoulis, K. Busch, and A. Lagendijk 323
Interference in Random Lasers
G. van Soest, F.J. Poelwijk, R. Sprik, and A. Lagendijk 324
Electron Patterns Under Bistable Electro-Optical
Absorption in Quantum Well Structures
C.A. Velasco, L.L. Bonilla, V.A. Kochelap, and V.N. Sokolov 325
Simulation of Mesoscopic Devices with Bohm Trajectories
and Wavepackets
X. Oriols, J.J. Garcia, F. Mart´ın, and J. Su˜n´e 327
Chaotic Motion of Space Charge Monopole Waves
in Semiconductors Under Time-Independent Voltage Bias
I.R. Cantalapiedra, M.J. Bergmann, S.W. Teitsworth, and L.L. Bonilla 329
Improving Electron Transport Simulation in Mesoscopic
Systems by Coupling a Classical Monte Carlo Algorithm
to a Wigner Function Solver
J. Garc´ıa-Garc´ıa, F. Mart´ın, X. Oriols, and J. Su˜n´e 330
Extended States
in Correlated-Disorder GaAs/AlGaAs Superlattices
V. Bellani, E. Diez, R. Hey, G.B Parravicini, L. Tarricone,
and F. Dom´ınguez-Adame 332

Non-Linear Charge Dynamics
in Semiconductor Superlattices
D. S´anchez, M. Moscoso, R. Aguado, G. Platero, and L.L. Bonilla 334
Time-Dependent Resonant Tunneling in the Presence
of an Electromagnetic Field
P. Orellana and F. Claro 336
XI I Contents
The Interplay of Chaos and Dissipation
in a Driven Double-Well Potential
S. Kohler, P. Hanggi, and T. Dittrich 337
Monte Carlo Simulation of Quantum Transport
in Semiconductors Using Wigner Paths
A. Bertoni, J. Garc´ıa-Garc´ıa, P. Bordone, R. Brunetti, and C. Jacoboni 338
Transient Currents Through Quantum Dots
J.A. Verg´es and E. Louis 340
Ultrafast Coherent Spectroscopy
of the Fermi Edge Singularity
D. Porras, J. Fern´andez-Rossier, and C. Tejedor 342
Self-Consistent Theory of Shot Noise Suppression
in Ballistic Conductors
O.M. Bulashenko, J.M. Rub´ı, and V.A. Kochelap 343
Transfer Matrix Formulation of Field-Assisted Tunneling
C. P´erez del Valle, S. Miret-Art´es, R. Lefebvre, and O. Atabek 345
Two-Dimensional Gunn Effect
L.L. Bonilla, R. Escobedo, and F.J. Higuera 346
An Explanation for Spikes in Current Oscillations
of Doped Superlattices
A. Perales, M. Moscoso, and L.L. Bonilla 347
Beyond the Static Aproximation in a Mean Field Quantum
Disordered System

F. Gonz´alez-Padilla and F. Ritort 349
Quantum-Classical Crossover of the Escape Rate
in a Spin System
X. Mart´ınez-Hidalgo 350
Appendix II List of Participants
Thermopower in Quantum Dots
K.A. Matveev
Department of Physics, Duke University, Durham, NC 27708-0305, USA
Abstract. At relatively high temperatures the electron transport in single elec-
tron transistors in the Coulomb blockade regime is dominated by the processes
of sequential tunneling. However, as the temperature is lowered the cotunneling
of electrons becomes the most important mechanism of transport. This does not
affect significantly the general behavior of the conductance as a function of the
gate voltage, which always shows a periodic sequence of sharp peaks. However, the
shape of the Coulomb blockade oscillations of the thermopower changes qualitati-
vely. Although the thermopower at any fixed gate voltage vanishes in the limit of
zero temperature, the amplitude of the oscillations remains of the order of 1/e.
1 Introduction
1.1 Coulomb Blockade
The phenomenon of Coulomb blockade is usually observed in devices where
the electrons tunnel in and out of a small conducting grain. A simplest ex-
ample of such a system is shown in Fig. 1. The small grain here is connected
to a large metal electrode—the lead—by a layer of insulator, which is so thin
that the electrons can tunnel through it.
When this happens, the grain acquires the charge of the electron −e.As
a result, the grain is now surrounded by an electric field, and there is clearly
some energy accumulated in this field. The energy can be found from classical
electrostatics as E
C
= e

2
/2C, where C is the appropriate capacitance of the
grain. Since the capacitance of small objects is small, the charging energy
can be quite significant. In a typical experiment E
C
/k
B
is on the order of
1 Kelvin. A typical temperature in this kind of experiment is T ∼ .1 K,
i.e., T  E
C
. Since it is impossible for an electron to tunnel into the grain
without charging it, the electron must have the energy E ≥ E
C
before it
tunnels. At low temperatures T  E
C
the number of such electrons in the
lead is negligible, and no tunneling is possible. This phenomenon is called the
Coulomb blockade of tunneling.
How can one observe the absence of tunneling? To do this, one needs
to add another metal electrode to the system—the gate, see Fig. 1. It is far
enough from the grain, so that no tunneling between these two pieces of metal
is possible. However by applying the voltage V
g
to the gate one can change
the charging energy and control the Coulomb blockade. Indeed, if we apply
positive voltage to the gate, the positive charge in it will attract electron to
the grain and decrease the charging gap. Mathematically, this is expressed
D. Reguera et al. (Eds.): Proceedings 1999, LNP 547, pp. 3−15, 1999.

 Springer-Verlag Berlin Heidelberg 1999
4 K.A. Matveev
+
V
g
Tunnel
junction
Small grain
Large lead Gate
C
g
C
l
+
+
+
+
+
+
+
+
Fig. 1. A small metallic grain is coupled to the lead electrode via a tunnel junction.
The electrostatic energy of the system is tuned by applying voltage V
g
to the gate
electrode. C
l
and C
g
are the capacitances between the grain and the lead and gate

electrodes.
as the following dependence of the electrostatic energy on the number n of
extra electrons in the grain and the gate voltage:
E(n, V
g
)=E
C

n −
C
g
V
g
e

2
. (1)
To discuss the effect of the gate voltage on electron tunneling in this system,
it is helpful to plot the energy (1) as a function of V
g
for various values of n,
see Fig. 2(a).
Clearly the energy (1) depends on V
g
quadratically, so for each value of n
we get a parabola centered at C
g
V
g
/e = n. If the number of electrons in the

grain can change due to the possibility of tunneling through the insulating
layer, the ground state of the system is given by the parabola with n being
the integer nearest to C
g
V
g
/e. Thus the number of the extra electrons in
the grain behaves according to Fig. 2(b). The steps of the grain charge as a
function of the gate voltage were observed by Lafarge et al. (1993).
Although the measurements of the charge of a small grain are possible, it
is far easier to measure transport properties of the systems with small metallic
conductors. The most common device studied experimentally is single elec-
tron transistor shown in Fig. 3. Unlike the device in Fig. 1, there are two leads
coupled to the grain by tunneling junctions. By applying bias voltage between
the two leads one can study the transport of electrons through the grain. In-
stead of making the device based on true metallic grains and leads one can
achieve the same basic setup by confining two-dimensional electrons in se-
miconductor heterostructures by additional gates, see, e.g., (Kastner 1993).
Thermopower in Quantum Dots 5
E
C
E
1
2
3
4
5
1234
n
C V /e

gg
(a)
(b)
n = n = n = n = 3210
G
(c)
C V /e
C V /e
gg
gg
1234
1234
Fig. 2. (a) Electrostatic energy (1) of the system in Fig. 1 as a function of the
charging energy for various values of the number of extra electrons n in the dot;
(b) the number of electrons in the dot as a function of the gate voltage found by
minimization of the electrostatic energy; (c) the conductance of a single electron
transistor shows peaks at the points where the charge has steps.
In this case the role of the grain is played by a small isolated “puddle” of
electrons—a quantum dot. Although there are significant differences between
these experimental techniques, they will not be important for the following
discussion.
An interesting behavior is observed when a small bias voltage is applied,
eV  T , and the conductance G of the single electron transistor is measured
as a function of the gate voltage. The experiment shows periodic peaks in
the conductance as a function of V
g
, see, e.g., (Kastner 1993).
The origin of the peaks is quite clear from Fig. 2(a). At the points where
C
g

V
g
/e = m +1/2, the electrostatic energy of the states with m and m +1
extra electrons in the grain are equal. At these values of the gate voltage an
electron can tunnel between the grain and the leads without changing the
6 K.A. Matveev
Left lead Right lead
Gate
V
g
Grain
(dot)
V
Fig. 3. Single electron transistor. The central electrode can be either a metal grain
or a semiconductor quantum dot. The bias voltage V is applied between the two
leads.
electrostatic energy of the system. As a result the Coulomb blockade is lifted,
and the transport is greatly enhanced. Thus the conductance has periodic
peaks, as shown in Fig. 2(c).
1.2 Mechanisms of Transport
Apart from the positions of the peaks in conductance of a single electron tran-
sistor, it is interesting to discuss their shapes. This requires a more detailed
understanding of the mechanisms of charge transfer through the grain. The
relative importance of different mechanisms is determined primarily by the
temperature. We will concentrate on the regime of temperatures much smal-
ler than E
C
, where the conductance does show the sharp peaks of Fig. 2(c).
In this case the two most important mechanisms are sequential tunneling and
cotunneling.

Sequential tunneling. This mechanism is the foundation of the so-called
orthodox model of Coulomb blockade (Averin and Likharev 1991). In order
for the current to flow from the left lead to the right one, one electron tunnels
from the left lead to the dot, and another electron tunnels from the dot to
the right lead. The two processes are assumed to be real transitions, so that
the energy of the system is conserved at every step. The resulting peak shape
was found by Glazman and Shekhter (1989):
Thermopower in Quantum Dots 7
G
sq
=
G
l
G
r
2(G
l
+ G
r
)
u/T
sinh(u/T )
. (2)
Here G
l
and G
r
are the conductances of the two tunnel barriers. The energy
u is the Coulomb blockade gap, which is proportional to the distance from a
peak, u =(eC

g
/C)(V
(n)
g
− V
g
), with V
(n)
g
=
e
C
g
(n −
1
2
) being the center of
the n-th peak, Fig. 2(c). The important features of the sequential tunneling
result (2) are:
– The peak height is
G
sq
0
=
G
l
G
r
2(G
l

+ G
r
)
. (3)
This result can be interpreted as the sum of the resistances of the two
tunneling barriers. The additional factor of 1/2 results from the fact that
near any given peak only two charge states n and n + 1 are allowed, and
all the tunneling events which would give rise to states with charged n−1
and n + 2 are forbidden.
– Away from the center of the peak the conductance falls off exponentially,
G
sq
∝ e
−u/T
. The reason for this behavior is that the electron tunneling
from a lead has to charge the grain, which requires for it to have the
energy u above the Fermi level, see Fig. 4(a). At low temperature T  u,
the probability of finding such an electron in the lead is exponentially
small.
E
F
u
(a) (b)
Fig. 4. Energy states of electrons in a single electron transistor. Quantum dot is
shown as a small region between the barriers separating it from the left and right
leads. Solid and dashed lines represent states below and above the Fermi level,
respectively. Arrows illustrate the elementary tunneling processes leading to (a)
sequential tunneling and (b) inelastic cotunneling.
Inelastic cotunneling. At low temperatures T  E
C

, the conductance
G
sq
in the valleys between peaks is exponentially small. As a result, another
8 K.A. Matveev
transport mechanism—the inelastic cotunneling—becomes important. This
mechanism is illustrated in Fig. 4(b). At the first stage, an electron tunnels
from a state near the Fermi level in the left lead to the dot. The energy of
the system increases by an amount close to u, assuming that we are not too
close to the center of the peak, i.e., u  T . Since the energy is not conserved,
the process does not stop here, and the state of the higher energy is only
a virtual state. At the second stage, another electron tunnels from the dot
to the right lead. This brings the energy back to its original value, and the
tunneling process is complete.
In the linear regime, when the bias is small, eV  T , the contribution of
inelastic cotunneling to the conductance was found by Averin and Nazarov
(1990):
G
co
=
π¯h
3e
2
G
l
G
r
T
2
u

2
. (4)
To compare this result with the sequential contribution, we need to estimate
G
co
at the center of the peak and in the valleys:
– At the center of a peak u = 0, and (4) formally diverges. This is because
the calculation was performed under the assumption u  T, and the
quasiparticle energies ξ ∼ T were neglected compared to u in the calcu-
lation of the energy of the virtual state. Thus the correct way to fix the
singularity in (4) is by substituting u ∼ T . Thus the peak value of G
co
is
G
co
0
∼
¯h
e
2
G
l
G
r
. (5)
– In the valleys, at u  T , the conductance is inversely proportional to
the square of the distance from the peak u. This result is easy to un-
derstand, because the amplitude of the second-order process is inversely
proportional to the energy of the virtual state E
v

 u, and the tunne-
ling probability is square of the amplitude. The temperature dependence
is T
2
, because the original electron of energy ξ ∼ T decays into three
quasiparticles, resulting in a phase space volume W ∝ ξ
2
∼ T
2
. This
argument is quite analogous to the one used to evaluate the lifetime of a
quasiparticle in a Fermi liquid, see, e.g, (Abrikosov 1988).
Comparison of the two mechanisms. At the center of the conductance
peaks one needs to compare the results (3) and (5). We are interested in
the case of weak tunneling between the quantum dot and the leads, i.e.,
G
l
+ G
r
 e
2
/¯h. Then obviously the sequential tunneling mechanism gives
the dominant contribution. On the other hand, the cotunneling conductance
(5) decays much slower than the sequential one, (3), when the gate voltage
is tuned away from the center of the peak. As a result, at u>u
c
, where
u
c
∼ T ln

e
2
¯h(G
l
+ G
r
)
, (6)
Thermopower in Quantum Dots 9
the cotunneling mechanism dominates the conduction. Note that this only
happens if u
c
is less than the distance to the center of the valleys u = E
C
.
Thus the cotunneling becomes an important mechanism of transport at low
enough temperatures T
<
∼
E
C
/ ln[e
2
/¯h(G
l
+ G
r
)].
It is worth mentioning that this crossover occurs far from the center of
the peak, i.e., u

c
 T, where the conductance is already very small. Thus
the presence of two transport mechanisms is not immediately obvious from
looking at the data for the conductance as a function of the gate voltage.
2 Thermopower
In a number of recent experiments a different transport property of single
electron transistors, the thermopower, was studied (Staring et al. 1993, Dzu-
rak et al. 1997). We will see below that the thermopower S is very sensitive
to the transport mechanism, and the crossover from sequential tunneling to
cotunneling changes the behavior of S(V
g
) qualitatively.
2.1 Definition
To measure the thermopower, one first needs to ensure that the temperatures
of the two leads T
l
and T
r
, are slightly different, ∆T = |T
l
−T
r
|T
l
. Then,
one must be able to measure the voltage V generated on the device under
the condition that there is no electric current I through it. The thermopower
is defined as
S ≡− lim
∆T →0

V
∆T




I=0
. (7)
It is helpful to think about the thermopower from the point of view of
the linear response theory. Most generally the current in a device is a linear
function of the voltage V and temperature difference ∆T , i.e.,
I = G
T
∆T + GV. (8)
Here G is the usual conductance of the system, and the kinetic coefficient
G
T
describes the current response to the temperature difference. Since the
definition of S calls for zero current I through he device, we can express the
thermopower (7) as
S =
G
T
G
. (9)
Thus, one can find the thermopower S by calculating the kinetic coefficients
G
T
and G.
10 K.A. Matveev

2.2 Physical Meaning of the Thermopower
Before we proceed with the discussion of the thermopower of single electron
transistors, let us try to get a better idea of the physical meaning of this
quantity.
The electric current in a rather arbitrary electronic device can be presen-
ted as
I = −e

[n
l
() −n
r
()]w()d, (10)
where  is the energy of an electron measured from the Fermi level, n
l
() and
n
r
() are the Fermi distribution functions corresponding to the temperatures
and chemical potentials of the left and right leads, respectively. The quantity
w() represents the remaining relevant physical properties of the system, such
as tunneling densities of states in the leads, transmission coefficients of the
tunneling barriers, etc.
Expression (10) is quite generic: it applies not only to simple tunne-
ling junctions, where w has the meaning of transmission coefficient, but also
to many other devices, including single electron transistors. If both the el-
ectrochemical potentials and temperatures in the two leads coincide, i.e.,
µ
l
− µ

r
= −eV = 0 and ∆T =0,wehaven
l
= n
r
and the current (10)
vanishes, as expected. One can then apply a small ∆T or V and discuss the
kinetic coefficients G
T
and G,
G
T
=
∂I
∂T
l
= −e


T

−
∂n
∂

w()d, (11)
G =
∂I
∂V
= −e


(−e)

−
∂n
∂

w()d. (12)
Note that we have differentiated only the Fermi function n
l
with respect to the
temperature and chemical potential of the left lead. Although the transmis-
sion probability w may also depend on T
l
and µ
l
, in the linear approximation
this should be neglected.
Using the results (11) and (12), we can present the thermopower (9) as
S = −

eT
. (13)
Here  has the meaning or the average energy of the electrons carrying the
current through the system. It is defined as
 =



−

∂n
∂

w()d


−
∂n
∂

w()d
(14)
We see from (13) that the thermopower S of a single electron transistor
measures the average energy of electrons tunneling between the left and right
leads.
It is worth mentioning that Π = −/e is the Peltier coefficient of the de-
vice, and that the relation S = Π/T equivalent to (13) follows from Onsager
relations, see, e.g. (Abrikosov 1988).
Thermopower in Quantum Dots 11
2.3 Thermopower in the Sequential Tunneling Regime
The first experiments on the thermopower of a single electron transistor (Sta-
ring 1993) were performed at relatively high temperature, and the trans-
port in the device was dominated by the sequential tunneling processes. The
theory of thermopower in this regime was developed by Beenakker and Sta-
ring (1992). At T  E
C
their results can be easily understood from Fig. 5.
E
F
E

F
S
V
g
E
C
E
C
u
Fig. 5. The thermopower of a single electron transistor as a function of the gate
voltage shows sawtooth behavior. This result was obtained within the framework
of the sequential tunneling theory by Beenakker and Staring (1992). The dashed
peaks correspond to the linear conductance G(V
g
).
We will interpret the result in terms of the average energy of tunneling
electrons (13). In the centers of the valleys separating the conductance pe-
aks the system possesses a certain symmetry: the change of the electrostatic
energy when one electron is either added to or removed from the dot is the
same, u = E
C
. As a result, the two processes shown in the left insert in Fig. 5
contribute equally to the transport, and the average energy of tunneling elec-
trons is zero. However, when the gate voltage is tuned slightly away from the
centers of the valleys, one of the processes gives much greater contribution to
the transport, resulting in  = ±E
C
. Thus the thermopower shows sharp
steps in the middles of the valleys of conductance. When the gate voltage
12 K.A. Matveev

is tuned away from the centers of the valleys, the change in the charging
energy u varies linearly with the gate voltage. One then expects  = u , and
S = −u/eT. In fact, the theory (Beenakker and Staring 1992) predicts
S
sq
= −
u
2eT
. (15)
The additional factor of
1
2
is due to the fact that in the sequential tunneling
mechanism the energy  of the tunneling electron can be less than u, if there
are holes in the dot at energy  − u. The density of electrons with energy 
in the lead is proportional to e
−/T
, and the density of holes at energy  −u
is e
−(u−)/T
. The product of these two exponentially small factors is simply
e
−u/T
, meaning that the tunneling probability is the same for all electrons
with energies  between 0 and u. The average energy of such electrons is then
 = u/2, in agreement with (15).
An important feature of the result (15) is that in the limit of low tempe-
rature, T → 0, the amplitude of the thermopower oscillations S
sq
0

= E
C
/2eT
diverges. This unusual behavior is specific to the sequential tunneling mecha-
nism. Unlike most other cases, the transport is due to electrons which are far
from the Fermi level, i.e., at energies  ∼ E
C
 T. Thus, according to (13)
the thermopower diverges as 1/T at T → 0.
The sawtooth behavior of the thermopower, Fig. 5, was observed experi-
mentally by Staring et al. (1993). The finite temperature of the experiment
gives rise to rounding of the “teeth” of the sawtooth dependence; the re-
lative positions of the peaks of conductance G(V
g
) to the sawtooth S(V
g
)
correspond to Fig. 5.
2.4 Cotunneling Thermopower
The problem of thermopower in single electron transistors has been recently
revisited in the experiment by Dzurak et al. (1997). Although the observed
behavior of S(V
g
) is somewhat similar to Fig. 5, there were a number of
important differences:
– The jumps aligned with the peaks of conductance, instead of the valleys.
– The behavior of S(V
g
) between the jumps was not linear.
– The direction of the “teeth” was opposite to the one shown in Fig. 5.

– The amplitude of the oscillations of S(V
g
) was estimated to be on the
order of S
0
∼ 1/e, i.e., much smaller than S
0
= E
C
/2eT .
In order to understand the deviations from the theory (Beenakker and Staring
1992), one needs to take into account the fact that the temperature in this
experiment was significantly lower than in (Staring et al. 1993). Indeed the
ratio T/E
C
in (Dzurak et al. 1997) was estimated to be on the order of
0.012, i.e., much less than 0.13 in (Staring et al. 1993). It is then natural to
conjecture that the new behavior observed by Dzurak et al. (1997) is caused
Thermopower in Quantum Dots 13
by cotunneling mechanism of transport, which is expected to dominate at
low temperatures, Sect. 1.2. Here we review the theory of the thermopower
in the regime of inelastic cotunneling (Turek and Matveev 1999).
Contrary to the case of sequential tunneling, the transport in the co-
tunneling regime is always due to the electrons which are within a strip of
width ∼ T around the Fermi level. Since the cotunneling occurs in the second
order of the perturbation theory, the cotunneling probability w is inversely
proportional to the square of the difference of energies of virtual and initial
states:
w() ∝
1

(u + 

− )
2
. (16)
Here  is the energy of the electron in the left lead, and 

is its energy after
it tunnels into the dot. It is clear from (16) that at positive u the electrons
above the Fermi level tunnel more effectively than those below the Fermi
level. Thus one expects to find non-zero average energy (14).
One can easily estimate  as follows. Since the typical  is of order T ,
one can expand (16) in small /u,
w() ∝
1
u
2

1+
2
u

. (17)
Thus although the electrons with positive  do tunnel more effectively than
the ones with negative , this effect is small as /u, or, for typical electrons,
∼ T/u. Since typical electrons have energies  ∼ T, the average energy is
∼T
2
/u. We can now use (13) to estimate the cotunneling thermopower
as S ∼−T/eu. A careful calculation supports this estimate and gives the

numerical prefactor:
S
co
= −
4π
2
5
T
e

1
u
+
1
u −2E
C

. (18)
The second term in (18) accounts for the processes when first an electron
tunnels from the dot to the right lead, and then another electron tunnels
from the left lead to the dot.
The cotunneling thermopower given by (18) diverges at u = 0. The origin
of this behavior is the same as that of divergence in cotunneling conduc-
tance result (4), namely the calculation at T  u neglects contributions of
quasiparticle energies to the energy of the virtual state. Taking this effect
into account, one can study the behavior of the thermopower at any u. This
leads to the smearing of the singularities at u → 0. In order to understand
the correct behavior of S(V
g
), one should also remember that at small u the

transport is dominated by sequential tunneling, Sect. 1.2. Thus both contri-
butions have to be taken into account in calculating G and G
T
in (9). The
resulting thermopower (Turek and Matveev 1999) is shown schematically in
Fig. 6. It is described by (18) in the valleys between the peaks of G(V
g
) and
coincides with sawtooth (15) in the peak regions.
14 K.A. Matveev
S
V
g
Fig. 6. Schematic view of the thermopower of a single electron transistor at low
temperatures. For comparison, the conductance peaks are shown by dashed line,
and the sawtooth behavior (15) is indicated by dash-dotted lines.
Note that the apparent slope of the new sawtooth is opposite to that of
the original one. It is also clear that the sharpest regions are now aligned with
the peaks of the conductance G(V
g
). To estimate the amplitude of the ther-
mopower oscillations, one can simply notice that the maxima are at u = u
c
,
where the crossover from sequential tunneling to cotunneling occurs. Substi-
tuting (6) into the sequential tunneling result, one arrives at the estimate of
the amplitude of the oscillations
S
0
∼

1
e
ln
e
2
¯h(G
l
+ G
r
)
. (19)
It is interesting that although at T → 0 and fixed gate voltage the thermo-
power vanishes in accordance with (18), the amplitude (19) is independent
of the temperature.
The behavior of Fig. 6 is in qualitative agreement with the experiment
(Dzurak et al. 1997). The exact amplitude of the thermopower oscillations
could not be measured in the experiment due to the uncertainty in measu-
rements of the temperatures of the leads. However, the order of magnitude
estimate of the amplitude of thermopower oscillations observed in that expe-
riment is in reasonable agreement with (19).
3 Conclusions
We discussed the thermopower of single electron transistors in the regime of
low temperatures, when sequential tunneling is no longer the main mecha-
nism of electron transport. We found that as the temperature is lowered and
inelastic cotunneling starts to dominate the conduction between the peaks of
Coulomb blockade, the dependence of the thermopower on the gate voltage
Thermopower in Quantum Dots 15
undergoes a qualitative change. This can be easily seen by comparing figures
5 and 6. The fact that the mechanism of transport can be clearly identified by
the general shape of S(V

g
) is new compared to the case of linear conductance
G(V
g
), which shows periodic peaks for either mechanism.
The results reviewed in this paper were obtained under the assumption
that the quantum dot is coupled weakly to the leads, i.e., the conductances of
the tunneling barriers are small compared to e
2
/¯h. In a recent experiment by
M¨oller et al. (1998) a different regime, in which one of the contacts is strongly
coupled to the lead, G
r
∼ e
2
/¯h, was investigated. The above theory is not
applicable to this case, however one can still explore the limit of almost perfect
transmission between the dot and one of the leads, when the conductance
G
r
approaches e
2
/π¯h. The results will be published elsewhere (Andreev and
Matveev 1999).
Another limitation of this work is that we have limited it to the regime of
relatively large dots or, equivalently, not too low temperatures. It is known
that in the limit T → 0 the transport will be dominated by elastic cotunneling
(Averin and Nazarov 1990). This happens at temperatures below
√
E

C
∆,
where ∆ is the quantum level spacing in the dot. Therefore, in small dots
one should expect that as the temperature is lowered the thermopower will
cross over from the sawtooth behavior of Fig. 5 to the inelastic cotunneling
dependence of Fig. 6, and then to a new regime of elastic cotunneling, which
needs to be studied in the future.
The author is grateful to A.V. Andreev, L.I. Glazman, and M. Turek for
useful discussions. This work was supported by A.P. Sloan Foundation and
by NSF Grant DMR-9974435.
References
Abrikosov A.A. (1988): Fundamentals of the theory of metals (Elsevier, Amsterdam)
Andreev A.V., Matveev K.A. (1999): in preparation.
Averin D.V., Likharev K.K. (1991): in Mesoscopic Phenomena in Solids, edited by
B. Altshuler, P.A. Lee, and R.A. Webb (Elsevier, Amsterdam)
Averin D.V., Nazarov Yu.V. (1990): Phys. Rev. Lett. 65, 2446
Beenakker C.W.J., Staring A.A.M (1992): Phys. Rev. B 46, 9667
Dzurak A.S., Smith C.G., Barnes C.H.W., Pepper M., Martin-Moreno L., Liang
C.T., Ritchie D.A., Jones G.A.C. (1997): Phys. Rev. B 55, 10197
Glazman L.I., Shekhter R.I. (1989): J. Phys. Conden. Matter 1, 5811
Kastner M.A. (1993): Physics Today 46,24
Lafarge P., Joyez P., Esteve D., Urbina C., Devoret M.H. (1993): Nature 365, 422
M¨oller S., Buhmann H., Godijn S.F., Molenkamp L.W., (1998): Phys. Rev. Lett.
81, 5197
Staring A.A.M., Molenkamp L.W., Alphenhaar B.W., van Houten H., Buyk O.J.A.,
Mabesoone M.A.A., Beenakker C.W.J., Foxon C.T. (1993): Europhys. Lett.
22,57
Turek M., Matveev K.A. (1999): in preparation.
Kondo Effect in Quantum Dots
L.I. Glazman

1
, F.W.J. Hekking
2
, and A.I. Larkin
1,3
1
Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455,
USA
2
Theoretische Physik III, Ruhr-Universit¨at Bochum, 44780 Bochum, Germany
3
L.D. Landau Institute for Theoretical Physics, 117940 Moscow, Russia
Abstract. Kondo effect in a quantum dot is discussed. In the standard Coulomb
blockade setting, tunneling between the dot and leads is weak, the number of elec-
trons in the dot is well-defined and discrete; Kondo effect may be considered in
the framework of the conventional one-level Anderson impurity model. It turns out
however, that the Kondo temperature T
K
in the case of weak tunneling is extremely
low. In the opposite case of almost reflectionless single-mode junctions connecting
the dot to the leads, the average charge of the dot is not discrete. Surprisingly, its
spin may remain quantized: s =1/2ors = 0, depending (periodically) on the gate
voltage. Such a “spin-charge separation” occurs because, unlike Anderson impurity,
quantum dot carries a broad-band, dense spectrum of discrete levels. In the doublet
state, Kondo effect with a significantly enhanced T
K
develops.
1 Introduction
The Kondo effect is one of the most studied and best understood problems
of many-body physics. Initially, the theory was developed to explain the in-

crease of resistivity of a bulk metal with magnetic impurities at low temper-
atures (Kondo 1964). Soon it was realized that Kondo’s mechanism works
not only for electron scattering, but also for tunneling through barriers with
magnetic impurities (Appelbaum 1966, Anderson 1966, Rowell 1969). A non-
perturbative theory of the Kondo effect has predicted that the cross-section
of scattering off a magnetic impurity in the bulk reaches the unitary limit
at zero temperature (Nozi`eres 1974). Similarly, the tunneling cross-section
should approach the unitary limit at low temperature and bias (Ng and Lee
1988, Glazman and Raikh 1988) in the Kondo regime.
The Kondo problem can be discussed in the framework of Anderson’s
impurity model (Anderson 1961). The three parameters defining this model
are: the on-site electron repulsion energy U, the one-electron on-site energy
ε
0
, and the level width Γ formed by hybridization of the discrete level with
the states in the bulk. The non-trivial behavior of the conductance occurs
if the level is singly occupied and the temperature T is below the Kondo
temperature T
K
 (UΓ)
1/2
exp{πε
0
(ε
0
+U)/2ΓU}, where ε
0
< 0 is measured
from the Fermi level (Haldane 1979).
It is hard to vary these parameters for a magnetic impurity embedded in

a host material. One has much more control over a quantum dot attached to
D. Reguera et al. (Eds.): Proceedings 1999, LNP 547, pp. 16−26, 1999.
 Springer-Verlag Berlin Heidelberg 1999
Kondo Effect in Quantum Dots 17
leads by two adjustable junctions. Here, the role of the on-site repulsion U
is played by the charging energy E
C
= e
2
/C, where C is the capacitance of
the dot. The energy ε
0
can be tuned by varying the voltage on a gate which
is capacitively coupled to the dot. In the interval
|N − (2n +1)| <
1
2
(1)
of the dimensionless gate voltage N, the energy ε
0
= E
C
[(2n+1)−N−1/2] <
0, and the number of electrons 2n + 1 on the dot is an odd integer. The level
width is proportional to the sum of conductances G = G
L
+G
R
of the left (L)
and right (R) dot-lead junctions, and can be estimated as Γ =(hG/8π

2
e
2
)∆,
where ∆ is the discrete energy level spacing in the dot.
The experimental search for a tunable Kondo effect brought positive re-
sults (Goldhaber-Gordon et al. 1998) only recently. In retrospect it is clear,
why such experiments were hard to perform. In the conventional Kondo
regime, the number of electrons on the dot must be an odd integer. How-
ever, the number of electrons is quantized only if the conductance is small,
G  e
2
/h, and the gate voltage N is away from half-integer values (see, e.g.,
Glazman and Matveev 1990, Matveev 1991). Thus, in the case of a quantum
dot, the magnitude of the negative exponent in the above formula for T
K
can be estimated as |πε
0
(ε
0
+ U)/2ΓU|∼(E
C
/∆)(e
2
/hG). Unlike an atom,
a quantum dot has a non-degenerate, dense set of discrete levels, ∆  E
C
.
Therefore, the negative exponent contains a product of two large parameters,
E

C
/∆ and e
2
/hG.
Further complication becomes evident if one compares the ∝ ln T correc-
tion G
K
, which is the textbook manifestation of the Kondo effect at T  T
K
,
with the background temperature-independent conductance G
el
provided by
the elastic co-tunneling mechanism,
G
K
∼ G
el
¯hG
e
2

∆
E
C

2
ln

E

C
T

. (2)
As one can see from Eq. (2), the Kondo correction remains small compared to
the background conductance everywhere in the temperature region T
>
∼
T
K
.
The Kondo contribution G
K
becomes of the order of e
2
/h and therefore
dominates the conductance only in the low-temperature region T
<
∼
T
K
. [The
ensemble-averaged value of G
el
at G
L
,G
R
 e
2

/¯h can be estimated (Averin
and Nazarov 1990) as G
el
(¯hG
L
G
R
/e
2
)(∆/E
C
).]
To bring T
K
within the reach of a modern low-temperature experiment,
one may try smaller quantum dots in order to decrease E
C
/∆; this route
obviously has technological limitations. Another, complementary option is
to increase the junction conductances, so that G
1,2
come close to 2e
2
/h,
which is the maximal conductance of a single-mode quantum point contact.
Junctions in the experiment (Goldhaber-Gordon et al. 1998, Cronenwett et al.
1998, Schmid 1998) were tuned to G  (0.3 −0.5)e
2
/π¯h. A clear evidence for
18 L.I. Glazman, F.W.J. Hekking, and A.I. Larkin

the Kondo effect was found at the gate voltages away from the very bottom
of the odd-number valley, where |ε
0
| is relatively small. Only in this domain
of gate voltages the anomalous increase of conductance G(T ) with lowering
the temperature T was clearly observed. (The unitary limit and saturation
of G signalling that T  T
K
, were not reached even there.) The anomalous
temperature dependence of the conductance, though, was hardly seen at N =
2n+ 1, where |ε
0
| reaches maximum. To increase the Kondo temperature and
to observe the anomaly of G(T ) function in these unfavorable conditions, one
may try to make the junction conductances larger. However, if G
1,2
come close
to e
2
/π¯h, the discreteness of the number of electrons on the dot is almost
completely washed out (Matveev 1995). Exercising this option, therefore,
raises a question about the nature of the Kondo effect in the absence of
charge quantization. It is the main question we address in this work.
2 Main Results
We show that the spin of a quantum dot may remain quantized even if charge
quantization is destroyed and the average charge Ne is not integer. Spin-
charge separation is possible because charge and spin excitations of the dot
are controlled by two very different energies: E
C
and ∆, respectively. The

charge varies linearly with the gate voltage, NN, if at least one of the
junctions is almost in the reflectionless regime, |r
L,R
|1, and its conduc-
tance G
L,R
≡ (2e
2
/h)(1 −|r
L,R
|
2
) is close to the conductance quantum. We
will show that the spin quantization is preserved if the reflection amplitudes
r
L,R
of the junctions satisfy the condition |r
L
|
2
|r
R
|
2>
∼
∆/E
C
. These two con-
straints on r
L,R

needed for spin-charge separation are clearly compatible at
∆/E
C
 1.
Under the condition of spin-charge separation, the spin state of the dot
remains singlet or doublet, depending on eN.IfcosπN < 0, the spin state
is doublet, and the Kondo effect develops at low temperatures T
<
∼
T
K
. The
Kondo temperature we find is
T
K
 ∆

∆
T
0
(N)
exp

−
T
0
(N)
∆

; (3)

T
0
(N)=αE
C
|r
L
|
2
|r
R
|
2
cos
2
πN. (4)
In the derivation presented below, we entirely disregard the mesoscopic fluc-
tuations. In this case, α>0 is some fixed numerical factor. Fluctuations
would result in a statistical distribution of α, with variance (δα)
2
∼α
2
.
Eps. (3) and (4) demonstrate that in the case of weak backscattering in the
junctions, the large parameter E
C
/∆ in the Kondo temperature exponent
may be compensated by a small factor ∝|r
L
|
2

|r
R
|
2
. This compensation, re-
sulting from quantum charge fluctuations in a dot with a dense spectrum
Kondo Effect in Quantum Dots 19
of discrete states, leads to an enhancement of the Kondo temperature com-
pared with the prediction for T
K
of a single-level Anderson impurity model,
discussed in the Introduction. Despite the modification of the Kondo temper-
ature, strong tunneling does not alter the universality class of the problem.
The temperature dependence of the conductance at T
<
∼
T
K
is described by a
known Costi et al. 1994 universal function F (T/T
K
),
G
K
(T/T
K
, N) 
e
2
h





r
R
r
L




2
(cos πN)
2
F (T/T
K
), (5)
with F(0)=1. Unlike the case of weak tunneling Ng and Lee 1988, Glazman
and Raikh 1988, the conductance (5) explicitly depends on the gate voltage.
Eqs. (3) – (5) were derived for an asymmetric set-up, |r
R
|
2
|r
L
|
2
.Inthe
special case |r

L
|→1, we can determine the energy T
0
, Eq. (4), exactly;
T
0
(N)=(4e
C
/π)E
C
|r
R
|
2
cos
2
πN, |r
L
|→1, (6)
where C =0.5772 is the Euler constant. The above results, apart from the
detailed dependence of T
K
and G
K
on N, remain qualitatively correct at
|r
L
|
2
|r

R
|
2
 1. The universality of the Kondo regime is preserved as long
as T
K
 ∆.
3 Bosonization for a Finite-Size Open Dot
We proceed by outlining the derivation of Eqs. (3)–(5). To see how the dense
spectum of discrete levels of the dot affects the renormalization of T
K
,we
first consider the special case |r
L
|→1 and |r
R
|1.
In the conventional constant-interaction model, the full Hamiltonian of
the system,
ˆ
H =
ˆ
H
F
+
ˆ
H
C
, consists of the free-electron part,
ˆ

H
F
=

dr

1
2m
∇ψ
†
∇ψ +(−µ + U(r)) ψ
†
ψ

, (7)
and of the charging energy
ˆ
H
C
=
E
C
2

ˆ
Q
e
−N

2

,
ˆ
Q
e
=

dot
drψ
†
ψ. (8)
Here the potential U(r) describes the confinement of electrons to the dot and
channels that form contacts to the bulk, µ is the electron chemical potential,
and operator
ˆ
Q is the total charge of the dot. To derive Eq. (3) for the Kondo
temperature, we start with a single-junction system. Following Matveev 1995,
we reduce the Hamiltonian (7) – (8) to the one-dimensional (1D) form, and
then use the boson representation for the electron degrees of freedom. In this
representation, the free-electron term is
ˆ
H
F
=
ˆ
H
0
+
ˆ
H
R

,